Rome, Flumicino, Italy

Taxi fare is a flat 48 euro (unless there have been recent increases). So your shuttle is in that ballpark.
The cheapest way is probably the bus (like Terravision), which can take you to Termini station for 6 euro pp.
https://www.adr.it/web/aeroporti-di-roma-en-/pax-fco-bus
The Leonardo Express train is 15 euro per person.
From the station to your hotel is nearly 1 mile walk.

From your hotel to the Vatican the fastest (and cheap) way is the Metropolitana (subway) Line A. Manzoni or Vittorio Emanuele are the closest stations to your hotel.

You might check and see if your hotel has a car service. We have done this now for the past few trips because we are just getting to a point in life that we don't want to try and catch public transportation after a long flight and with luggage. And the same getting back to the airport. I have found that the taxi fares never seem to be exactly what they say the flat rate it. Always costs us a bit more. If you are traveling with others the car service is very economical split with another person or couple. We have always had reliable service with a hotel car service, or a suggestion from your hotel.

For 14 euro you can take the Leonardo Express from Fiumicino to Roma Termini. From there is looks like about a 10 minute walk to your hotel.

I would take a carry-on size bag and a small bag (back pack, messenger bag) to carry snacks, water, etc, around with you.

Mid-size for me is just way to big.

Best to travel with a small wheeled 20 inch carry on bag per person. Total weight should be under maximum allowed on the flight, and only what you can manage even after a long day of travel. For me that's 15 pounds total weight.

Backpacks may fall into the hand bag category which also has an onboard weight limit. Best to pack it well under that weight. Don't know your plans in Italy, but I have witnessed backpacks sabotage efforts to climb stairs to a train car while they were trying to get their luggage up at the same time.

Surely you don't plan to carry a backpack around all day with you. I take my RS Euro flight bag only when I want my travel laptop with me, but there's no point to carry that bag around all day. I leave the laptop locked in my luggage in my hotel room. I walk around only with my RS neck wallet and RS money belt.

Best not to add any weight on you while you are sightseeing. Keep it light and as close to being at home as you can.

When OP asks for best & most economical way to Rome, a car service is probably the best and the least economical way. Best and most economical way are general not compatible words. The most economical way is going to be the bus (8 Euro) to Termini and walk from there are a very short cab ride. As mentioned, door to door taxi is 48E for up to four people and luggage. We have never had a problem with it not being 48E but there is an extra for very late at night or early morning. Between the taxi or the bus, you can take the Leonard Express at 15/person or the FM1 (I think) at about 8E to Termini and again, walk or taxi from there.

When you ask about a backpack plus a mid-side suitcase, it has little meaning since no one knows how big each is. Full external frame backpack is completely out. Too inconvenient, too big. And, to me, a mid-side suitcase sounds too big. A carryon size bag that also has straps that can be worn as a back pack is more reasonable. For many people, some exceptions, a one carryon side bag, either rolling or backpack style, is more than adequate.

If you want better answers, then describe the size of your bags. Anything more than 10x12x20 give or take a couple inches is too big.

Ronald. Some people like myself do live dangerously and take a mid sized case. Despite the apocalyptic warnings of the RS carry on militia, we make it out alive and even manage to enjoy our trip (perhaps because we are not spending every second night washing underwear in a bathroom sink with a tube of toothpaste).

Join the dark side and say no to those whose lives are defined by 20 inches.